Questions tagged [spherical-geometry]
geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect
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Idealized sailing East along a latitude in absence of crosswind and crosscurrent
Assume a spherical Earth.
My boat's axis is aligned East-West. Rudder is aligned with boat axis. Will I sail along the latitude ?
My doubt is caused by the following observations: the latitude makes a ...
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Can you find a great circle with only a compass?
Thinking about the construction of temari balls got me thinking about how one might begin the marking portion of the process, particularly how to build a great circle in the first place. To make the ...
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Calculating the intercept course/point of two objects on the surface of a sphere - target moving on a rhumb line, interceptor on a great circle
My question is very similar to How to find the launch direction to intercept an object moving on a sphere? But the answer there assumes both objects move on great circles. My problem involves a target ...
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How do I find the point of latitude on a meridian that is closest to a specified point?
I need to find the latitude of the point on a meridian that has the shortest great-circle distance to a general but specified point elsewhere on the globe.
To find out I might (in theory) do some ...
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Spherical excess in four dimensions
I would like to understand the tesselation of the four dimensional space by 24 cells from the point of view of spherical excess. Therefore, I tried to lift the ideas from three dimensions to four ...
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Intrinsic formula for calcuating latitude and longitude for a fraction of a great circle.
This is not a homework problem; my two degrees are in engineering. My interest in solving this problem stems from a broader amateur research project. Let $\phi$ be latitude and $\lambda$ longitude. ...
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calculation discrepancy measuring arclength between points in Cartesian vs. spherical coordinates
I am calculating the length of an arc between two points on a sphere labeled in both Cartesian coordinates and spherical coordinates to confirm that my conversions and formulas are correct. I'm ...
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deriving simplified haversine formula from standard version
On the wikipedia page on haversines:
https://en.wikipedia.org/wiki/Haversine_formula
It gives the following formula for the distances between any two points given their latitudes and longitudes (I ...
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Is Girard's Theorem always true for concave spherical polygons?
It appears that the answer is yes, that one can always split a concave spherical polygon into adjacent spherical triangle from at least one vertex on the polygon, like in the example below:
But I was ...
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Is there a generalization of the term "spherical lune" for the region of a sphere bounded by arcs of two not-necessarily-great circles?
The definition of a spherical lune is "the shape formed by two great circles and bounded by two great semicircles which meet at their antipodes". However, I haven't been able to find a term ...
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How to find the central angles in a spherical pyramid given its spherical triangle base?
Given $\alpha$, $\beta$, and $\gamma$, or even just knowing the excess of the spherical triangle, is there a way to calculate the central angles $\angle AOB$, $\angle AOC$, and $\angle BOC$ in order ...
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Sphere average of the second symmetric sum of a matrix
Say given $B$ symmetric and tr $B=0$. Let $P_u$ be the projection onto the tangent plane attached to $u$, i.e., $ P_u=Id-u\otimes u$. Let $\sigma_2(M)$ be the second symmetric sum of eigenvalues of $...
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Is it possible to write a set of 2D functions that are each positive for only one section of this divided sphere?
I have a unit sphere divided into 14 patches: 6 identical square-ish sections and 8 identical triangle-ish sections, with the arrangement and all of the symmetries of a cuboctahedron. The boundaries ...
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Maximum area coverage of the sphere by 5 caps
Place 5 congruent spherical caps (geodesic balls) on the unit sphere such that no four or more of the caps' centers are coplanar. Must there exist a maximal area configuration of such caps in which ...
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3D Volume Enclosed by a Spherical Cap and a Point on an Inner Sphere
I have two concentric spheres:
Inner sphere with radius r₁
Outer sphere with radius r₂, where r₁ < r₂
On the surface of the outer sphere, there is a spherical triangle (defined by three geodesic ...
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